For a finite commutative ring R, let a,b,c∈R be fixed elements. Consider the equation ax+by=cz where x, y, and z are idempotents, units, any element in respectively. We say that R satisfies r-monochromatic clean condition if, for r-colouring χ of elements there exist x,y,z∈R with χ(x)=χ(y)=χ(z) such holds. define m(a,b,c)(R) to least positive integer r does not satisfy condition. This means exi...

The concept of Lucky k-polynomials and in particular Lucky χ-polynomials was recently introduced. This paper introduces Stirling number of the fourth kind and Lucky partitions of a finite set in order to determine either the Lucky k- or Lucky χ-polynomial of a graph. The integer partitions influence Stirling partitions of the second kind.

A set of problems is offered to accompany the set of lectures with the above title. Topic 1: Coherent States 1.1 Let {|n〉}, n = 0, 1, 2, 3, . . ., denote a complete set of orthonormal vectors in an abstract Hilbert space , with 〈m|n〉 = δmn. Next, let |χ〉 = √ 1 − |χ|2 |0〉 + χ|1〉 , for all complex χ, 0 ≤ |χ| ≤ 1, denote a family of two-dimensional vectors. a) If χ→ χ in the sense of complex numbe...

(1) (c, χ) = (7n, n), (2) (c, χ) = (5n, n), (3) (c, χ) = (4n, n), (4) (c, χ) = (2n, n), (5) (c, χ) = ((6 + 8g)n, (1 + g)n (for g ≥ 0), (6) (c, χ) = (7n+ (6 + 8g)m,n+ (1 + g)m), (7) (c, χ) = (7n+ 5m,n+m), (8) (c, χ) = (7n+ 4m,n+m), (9) (c, χ) = (7n+ 2m,n+m), (10) (c, χ) = ((6 + 8g)n+ 5m, (1 + g)n+m) (for g ≥ 0), (11) (c, χ) = ((6 + 8g)n+ 4m, (1 + g)n+m) (for g ≥ 0), (12) (c, χ) = ((6 + 8g)n+ 2m,...

The expected trilepton signals from p¯ p → χ ± 1 χ 0 2 → (χ 0 1 ℓ ± ν)(χ 0 1 ℓ ′+ ℓ ′−) will be converted into hadronically quiet multilepton signals, if the two final χ 0 1 have leptonic R-parity-violating (RPV) decays χ 0 1 → ℓℓ ′ ν. We make illustrative calculations of the acceptance for these spectacular RPV signals, and point out that distinctive multilepton signals are possible even when ...

Let L(s,χ) be the Dirichlet L-function associated to a non-principal primitive character χ defined modq, where q≥3. We prove, under assumption of Generalised Riemann Hypothesis, validity estimates given by Lamzouri, Li, and Soundararajan on |L(1,χ)|. As corollary, we have that similar hold for class number imaginary quadratic field Q(−q), q≥5.

The multipartite entangled state n p χ χ χ ,..., , , 3 2 for the total momentum and relative coordinates of n particles is constructed. The corresponding quantum mechanical operator with respect to the classical transformation p p e 1 λ → , i i i e χ χ λ → , n i ,..., 2 = , in the state n p χ χ χ ,..., , , 3 2 is investigated.